A procedure is demonstrated for determining the Runge-Lenz vector for the non-relativistic, classical Kepler problem. This vector constant of the motion enables one to determine the orbit of the particle in a simple way. It is also useful in finding the quantum mechanical solutions for this problem and for systematically studying the symmetry characteristics involved. This article, however, stresses only the classical aspects. The method used can, with appropriate modification, be applied to other problems.